Problem 45 - Intermediate Algebra

Question

Calculate how many times more intense an earthquake with a Richter number of \(7.3\) is than an earthquake with a Richter number of \(6.4\).

Step-by-Step Solution

Answer

The earthquake with a Richter number of 7.3 is approximately 7.943 times more intense than one with 6.4.

Step 1: Understand the Richter Scale

The Richter scale measures the magnitude of an earthquake on a logarithmic scale. That means each whole number increase on the scale represents a tenfold increase in measured amplitude and roughly 31.6 times more energy release.

Step 2: Formula to compare Intensity

To find out how many times more intense one earthquake is than another, use the formula: \[ I = 10^{(M1 - M2)}\]where \( I \) is the intensity ratio, and \( M1 \) and \( M2 \) are the magnitudes of the earthquakes.

Step 3: Substitute the Richter Numbers

Substitute the given Richter numbers into the formula. Here, \( M1 = 7.3 \) and \( M2 = 6.4 \). Therefore, \[I = 10^{(7.3 - 6.4)}\]

Step 4: Calculate the Difference

Calculate the difference in magnitude between the two earthquakes. \[7.3 - 6.4 = 0.9\]

Step 5: Calculate the Intensity Ratio

Use the result from Step 4 in the formula from Step 2. \[I = 10^{0.9}\]Calculate the value: \[I \approx 7.943\]Thus, an earthquake with a Richter number of 7.3 is approximately 7.943 times more intense than an earthquake with a Richter number of 6.4.

Key Concepts

Logarithmic Scale

The Richter scale is a fascinating method used to measure the magnitude of earthquakes. It operates on what is known as a logarithmic scale. This might sound complicated, but it's actually a clever way to handle extremely large variations in data, such as earthquake magnitudes. When we say the Richter scale is logarithmic, it means that each unit increase on the scale represents a tenfold increase in amplitude.
This also corresponds to about 31.6 times more energy being released by the earthquake. This is distinct from a linear scale, where each unit increase would add a constant amount. In the case of earthquakes, where the magnitude can vary drastically, using a logarithmic scale helps compress the range into a more manageable format.

• Logarithmic scales simplify comparisons between large numbers.
• They make it easier to display and interpret large magnitude differences.
• Each increment in magnitude represents exponentially greater intensities.

This scale allows us to understand the relative strength of earthquakes with just a simple number.

Earthquake Intensity

Understanding earthquake intensity is crucial when discussing the Richter scale. Intensity refers to the strength or force of an earthquake as it is felt or measured. On the Richter scale, each increase in number signifies a significant jump in explosive force.
For instance, an earthquake with a magnitude of 5.0 releases much less energy than a 6.0-magnitude shock. As we learned, every increase by one on the Richter scale means about 31.6 times more energy release.

• Intensity conveys the destructive power of an earthquake.
• It shows how much an area is affected depending on the quake's magnitude.
• The logarithmic nature exacerbates how these intensities can differ dramatically.

Calculating intensity using the Richter scale allows us to precisely determine how much more powerful one earthquake is compared to another, as seen in our previous exercise.

Magnitude Comparison

Comparing magnitudes of earthquakes is straightforward yet powerful due to the logarithmic nature of the Richter scale. Let's say we want to compare two earthquakes with magnitudes 7.3 and 6.4. By using the formula for intensity comparison, we capture this difference.To compare, we take the difference in magnitudes: \[ M_1 - M_2 = 7.3 - 6.4 = 0.9 \] Using the formula \[ I = 10^{(M_1 - M_2)} \] we find the intensity ratio is \[ I = 10^{0.9} \approx 7.943 \].

• This means a magnitude 7.3 earthquake is approximately 7.943 times more intense than a magnitude 6.4 earthquake.
• Understanding how to compute these ratios reveals just how dramatically different two similarly ranked earthquakes can truly be.

Thus, magnitude comparison helps us grasp the potential impact of one quake over another effectively.